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The paper investigates the existence of global attractor for a strongly damped wave equation with fully supercritical nonlinearities: $ u_{tt}-Δ u- Δu_t+h(u_t)+g(u)=f(x) $. In the case when the nonlinearities $ h(u_t) $ and $ g(u) $ are of fully supercritical growth, which leads to that the weak solutions of the equation lose their uniqueness, by introducing the notion of limit solutions and using the theory on the attractor of the generalized semiflow, we establish the existence of global attractor for the subclass of limit solutions of the equation in natural energy space in the sense of strong topology.

In this paper, we are concerned with the existence and stability of pullback exponential attractors for a non-autonomous dynamical system. (ⅰ) We propose two new criteria for the discrete dynamical system and continuous one, respectively. (ⅱ) By applying the criteria to the non-autonomous Kirchhoff wave models with structural damping and supercritical nonlinearity we construct a family of pullback exponential attractors which are stable with respect to perturbations.

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